374 THE THEORY OF THE HELMHOLTZ RESONATOR [
and the third is, as for the principal term,
Z) I 7-v r*r\Ci XTV
.(51
0 + b cos
Thus altogether, when the three integrals are taken between the Hi: 0 and 6l, we get
- 6 cos V(a2 - 52) + [K + & (2 cos2 (j> + £ sin2 </> - 1)]
- sn2
X ~
and finally after integration with respect to
Thus altogether the integral on the left of (42) becomes
3 9
.(53)
In consequence of the occurrence of b-, this expression cannot be m to vanish at all points of the aperture, a sign that the assumed form o: is imperfect. If, however, we neglect the last term, arising from — 6 2, (in) — S (out), our expression vanishes provided
3 9 .. 27r
showing that a is of the order k"c", so that this equation gives the reltv between a and kc to a sufficient approximation. Helmholtz's solution co spends to the neglect of the second and third terms on the left of (54), mill
_3_ _ 27T _ 27TC
k-c~ a R ' ........................... ^' *"
where R denotes the linear radius of the circular aperture. If we introc A- (=
S denoting the capacity of the sphere, the known approximate value.
The third term on the left of (54) represents the decay of the vibra due to the propagation of energy away from the resonator. Omitting for the moment, we have as the corrected value of X,
Let us now consider the term representing decay of the vibrations. time factor, hitherto omitted, is eihvt, or if we take k = k,+ikz, e~k*n e If t = r, the period, fc1Fr = 27r, and e~^Vr = e-***/*'. This is the factoi which the amplitude of vibration is reduced "in one period. Now from (5
j kc =
37T2
- — ,
27TC/ '
* ru ^ j 7T
[i?or — read -— , and three lines below read
" arising from - §g* in sin 6 [2 (in) - S (out)] " :_ see footnote on p. 372. W. F. S.] ($ H- b cos c6)2} a2 — b* sin2 <jf> . ^ #-t-&cos</> ( Q.